Question

If a pair of opposite angles of a quadrilateral are supplementary, the quadrilateral is:

• A. cyclic
• B. parallelogram
• C. rectangle
• D. rhombus

Answer 1

Answer: (A)

cyclic

Let $ABCD$ be a cyclic quadrilateral.

Join $OB$ and $OD.$

$\Rightarrow$ $\angle BOD=2\angle BAD$

$\Rightarrow$ $\angle BAD=\frac{1}{2}\angle BOD$

Similarly, $\angle BCD=\frac{1}{2}\angle DOB$

$\Rightarrow$ $\angle BAD+\angle BCD=\frac{1}{2}\angle BOD+\frac{1}{2}\angle DOB$

$=\frac{1}{2}(\angle BOD+\angle DOB)$

$=\frac{1}{2}\times {360}^o$

$={180}^o.$

$\therefore$ $\angle A+\angle C={180}^o$

Similarly, $\angle B+\angle D={180}^o.$

$\therefore$ If a pair of opposite angles of a quadrilateral are supplementary, the quadrilateral is a $cyclic.$